https://file.aiquickdraw.com/mathbot/file/1726707674167fp1f7gty.jpg

The expression to simplify is:

\[
E(p_1, p_2, \bar{U}) = \frac{\bar{U} p_1^{1/5} p_2^{4/5}}{4^{4/5}} + \frac{\bar{U}^{4/5} p_2^{4/5} p_1^{1/5}}{1}
\]

Let's break this down and simplify it:

### Step 1: Identify common factors
We observe that both terms in the expression share some common elements:
- \(\bar{U}\) (or \(\bar{U}^{4/5}\))
- \(p_1^{1/5}\)
- \(p_2^{4/5}\)

Thus, we can factor out these common terms.

\[
E(p_1, p_2, \bar{U}) = p_1^{1/5} p_2^{4/5} \left( \frac{\bar{U}}{4^{4/5}} + \bar{U}^{4/5} \right)
\]

### Step 2: Combine terms inside the parentheses
The expression now simplifies to:

\[
E(p_1, p_2, \bar{U}) = p_1^{1/5} p_2^{4/5} \left( \frac{\bar{U}}{4^{4/5}} + \bar{U}^{4/5} \right)
\]

We leave this as the simplified form unless there is a further requirement to manipulate the expression inside the parentheses.

This is the simplified form of the given equation. Would you like further details on any part of this process, or do you have more questions?

### Related Questions:
1. How do you factor common terms in more complex expressions?
2. What are the properties of exponents that allow us to simplify expressions like this?
3. How can we handle fractional exponents in general expressions?
4. What is the significance of the utility function \(\bar{U}\) in economics?
5. How would we approach simplifying if we added more terms or factors?

### Tip:
When simplifying expressions with common factors, always start by identifying shared terms across all components to streamline the process.

Simplify: E(p1, p2, Ū) = (Ū * p1^(1/5) * p2^(4/5)) / 4^(4/5) + (Ū^(4/5) * p2^(4/5) * p1^(1/5)) / 1

Simplify the expression involving exponents and factoring: E(p1, p2, Ū) = (Ū * p1^(1/5) * p2^(4/5)) / 4^(4/5) + (Ū^(4/5) * p2^(4/5) * p1^(1/5)) / 1. This step-by-step solution covers exponent rules and algebraic simplification. Suitable for Grades 11-12 and college-level math.

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